BME 6938Neurodynamics

Instructor: Dr Sachin S Talathi

Cable Equation: Transient Solution

Green’s Function G(X,T) for infinite cable: solution of above equation for:

With initial condition: and Boundary condition:

General Solution to Cable Equation:

Hint: Use the formula:

Graphical representation

Two points worth noticing:1.The potential is described as a Gaussian function centered at the site of current injection that broadens and shrinks in amplitude with time2. Membrane potential measured from further location reaches its maximum value at later time

Ralls Model-Equivalent cylinder

Practical Situation: Choose L=average eletrotonic length of all dendritic branches and

Also read the classic papers by Rall (posted on the web: Rall_1959, Rall_1962, Rall_1973)

Ralls Assumptions:1.All membrane properties are the same2.All terminal branches end with same boundry condition3.All terminal branches end at same electrotonic distance from soma4. At every branch point 3/2 rule is obeyed5. Any dendritic input must be delivered proportionally to all branches at a given electrotonic distance

Synaptic IntegrationModel for current injection into neuron through synapse-alpha function

Imp Points to Note:1.Distal synaptic input produces measurable signals in the soma2.Obvious fact, the measured EPSP at soma due to stimulation at distal input is smaller as compared to closer to soma3.Rise time to peaks is progressively delayed for inputs at increasing distance from soma4.Decay times of all the inputs is the same, ie. Potential cannot decay slower than membrane time constant

We know now that dendrites are not passive, what is the current view on dendritic integration. Read the 2 review articles (Magee_2000, London_2005) posted on the website

Nonlinear Membrane:

Ions: Na+, K+, Ca2+, Cl-

Re-visiting Goldman Equation: The idea of time dependent conductance

Membrane voltage and time dependent ion channel conductance

First order approximation

Na+ reversal potential (Nernst Equilibrium Potential)

The Gate Model HH proposed the gate model to provide a quantitative framework

for determining the time and membrane potential dependent properties of ion channel conductance.

The Assumptions in the Gate Model: Membrane comprise of aqueous pores through which the ions

flow down their concentration gradient These pores contain voltage sensitive gates that close and open

dependent on trans membrane potential The transition from closed to open state and vice-versa follow

first order kinetics with rate constants: and

I have posted the original paper by HH where they proposed to above model and the experiments they conducted to develop the now famous HH model for neuron membrane dynamics (HH_1952). I urge all of you to read this classic work; that led them to receive Nobel Prize in Medicine (The only Noble Prize so far that the field of computational neuroscience have produced)

Kinetics of Gate Transition Let P represent the fraction of gates within the

ion channel that are open at any given instant in time

1-P then represents the fraction that are closes at that instant in time

If and are the rate constants we have

Open P

Closed 1-P

Steady State: Steady state implies

Multiple Gates: If a ion channel is comprised of multiple gates;

then each and every gate must be open for the channel to conduct ion flow.

The probability of gate opening then is given by:

Gate Classification Activation Gate: P(t,V) increases with membrane

depolarization Inactivation Gate: P(t,V) decreases with

membrane depolarization

The unknowns In order to use the gate model to describe

channel dynamics of cell membrane HH had to determine the following 3 things Macro characteristics of channel type The number and type of gates on each channel The dependence of transition rate constants

and on membrane voltage V

The Experiments Two important factors permitted HH analysis as they

set about to design experiments to find the unknowns Giant Squid Axon (Diameter approx 0.5 mm), allowed for

the use of crude electronics of 1950’s (Squid axon’s utility for of nerve properties is credited to J.Z Young (1936) )

Development of feed back control device called the voltage clamp capable of holding the membrane potential to a desired value

Before we look into the experiments; lets have a look at

the model proposed by HH to describe the dynamics of squid axon cell membrane

The HH model HH proposed the parallel conductance model

wherein the membrane current is divided up into four separate contributions Current carried by sodium ions Current carried by potassium ions Current carried by other ions (mainly chloride and

designated as leak currents) The capacitive current

We have already seen this idea being utilized in GHK equations

The Equivalent Circuit

Current flows assumes Ohms Law

Goal: Find and

Example: Time and Voltage dependence of Potassium channel conductance

The Experiments

Space Clamp: Eliminate axial dependence of membrane voltage Stimulate along the entire length of the axon Can be done using a pair of electrodes as shown Provides complete axial symmetry

Result:Eliminate the axial component inThe cable equation

Voltage Clamp: Eliminate Capacitive Current

http://www.sinauer.com/neuroscience4e/animations3.1.html

Example of Voltage Clamp Recording

Clamped Voltage=20 mV

The sum of parts

Note the different time scale

Series of Voltage Clamp Expt

Selective blocking with pharmacological agents

TTX: Tetrodotoxin; selectively blocks sodium channels TEA: Tetraethylamonium; selectively blocks potassium channels

H-H experiments to test Ohms law

Foundation of Cellular Neurophysiology, Johnston and Wu

HH measurement of Na and K conductance

Gating variables

Maximum conductance

Functional fitting to gate variable We see from last slide Na comprise of activation and inactivation K comprise of only activation term HH fit the the time dependent components of

the conductance such that

Activation gate Inactivation gate

m,n and h are gate variables and follow first order kinetics of the gate model

Gate model for m,n and h

Activation: Inactivation:

Determine and Use the following relationship

Do empirical curve fitting to obtain

Determining and

Profiles of fitted transition functions

Summary of HH experiments Determine the contributions to cell membrane

current from constituent ionic components Determine whether Ohms law can be applied

to determine conductances Determine time and voltage dependence of

sodium and potassium conductances Use gate model to fit gating variables Use equations from gate model to determine

the voltage dependent transition rates

The complete HH model